Properties

Label 2-8004-1.1-c1-0-34
Degree $2$
Conductor $8004$
Sign $1$
Analytic cond. $63.9122$
Root an. cond. $7.99451$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2.25·5-s + 3.02·7-s + 9-s − 4.27·11-s + 1.71·13-s − 2.25·15-s + 2.42·17-s + 1.07·19-s + 3.02·21-s − 23-s + 0.0730·25-s + 27-s + 29-s + 9.45·31-s − 4.27·33-s − 6.80·35-s + 7.34·37-s + 1.71·39-s − 3.99·41-s − 6.28·43-s − 2.25·45-s + 0.0164·47-s + 2.13·49-s + 2.42·51-s − 0.890·53-s + 9.62·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.00·5-s + 1.14·7-s + 0.333·9-s − 1.28·11-s + 0.475·13-s − 0.581·15-s + 0.587·17-s + 0.245·19-s + 0.659·21-s − 0.208·23-s + 0.0146·25-s + 0.192·27-s + 0.185·29-s + 1.69·31-s − 0.744·33-s − 1.15·35-s + 1.20·37-s + 0.274·39-s − 0.623·41-s − 0.958·43-s − 0.335·45-s + 0.00239·47-s + 0.305·49-s + 0.339·51-s − 0.122·53-s + 1.29·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8004\)    =    \(2^{2} \cdot 3 \cdot 23 \cdot 29\)
Sign: $1$
Analytic conductor: \(63.9122\)
Root analytic conductor: \(7.99451\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8004,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.272788196\)
\(L(\frac12)\) \(\approx\) \(2.272788196\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
23 \( 1 + T \)
29 \( 1 - T \)
good5 \( 1 + 2.25T + 5T^{2} \)
7 \( 1 - 3.02T + 7T^{2} \)
11 \( 1 + 4.27T + 11T^{2} \)
13 \( 1 - 1.71T + 13T^{2} \)
17 \( 1 - 2.42T + 17T^{2} \)
19 \( 1 - 1.07T + 19T^{2} \)
31 \( 1 - 9.45T + 31T^{2} \)
37 \( 1 - 7.34T + 37T^{2} \)
41 \( 1 + 3.99T + 41T^{2} \)
43 \( 1 + 6.28T + 43T^{2} \)
47 \( 1 - 0.0164T + 47T^{2} \)
53 \( 1 + 0.890T + 53T^{2} \)
59 \( 1 + 7.48T + 59T^{2} \)
61 \( 1 - 13.8T + 61T^{2} \)
67 \( 1 + 1.64T + 67T^{2} \)
71 \( 1 - 1.92T + 71T^{2} \)
73 \( 1 + 11.8T + 73T^{2} \)
79 \( 1 + 6.09T + 79T^{2} \)
83 \( 1 - 10.8T + 83T^{2} \)
89 \( 1 - 0.796T + 89T^{2} \)
97 \( 1 - 17.5T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.953535639628357198380382257654, −7.53237024005889941478701889212, −6.56829331661125957856764655243, −5.63197949339279567298658398340, −4.83879833798524886907432814434, −4.36598992950434239194454538522, −3.44949112432146472413271320171, −2.77876057721605905617644008168, −1.81198296217733075029631405198, −0.73578591850459176488466117041, 0.73578591850459176488466117041, 1.81198296217733075029631405198, 2.77876057721605905617644008168, 3.44949112432146472413271320171, 4.36598992950434239194454538522, 4.83879833798524886907432814434, 5.63197949339279567298658398340, 6.56829331661125957856764655243, 7.53237024005889941478701889212, 7.953535639628357198380382257654

Graph of the $Z$-function along the critical line